Getting Started with Individual Differences Research

A Practical Introduction for Experimental Researchers

Jason Tsukahara, Ph.D.

University of Miami

December 9, 2025

Workshop Overview

Why individual differences matter
Taking a latent variable approach
Hands-on practice in R

The Experimental Approach

Manipulate variables → measure change
Control sources of variability
Focus on mean differences between conditions
Eliminate variability between individuals

The individual differences approach

Measure variables → examine covariation across people
Embrace variability between individuals as meaningful
Focus on relationships between constructs
Eliminate variability within individual

Two ways of thinking about variability

Experimental

Variability within individuals = Signal
Variability between individuals = Noise
Focus: Mean differences
Question: Does x cause y?

Individual Differences

Variability within individuals = Noise
Variability between individuals = Signal
Focus: Correlations / Covariance
Question: Who differs and why?

Mean differences tell us whether an effect exists, individual differences tells us for who and why

Variability between individuals is not just noise

Visual working memory as an example

Visual working memory capacity is measured with change-detection “visual arrays” tasks
Behavioral and neural evidence from this task suggests that the capacity-limit of visual working memory is about 3-4 items

Variability between individuals is not just noise

Visual working memory as an example

Visual working memory capacity is measured with change-detection “visual arrays” tasks
Behavioral and neural evidence from this task suggests that the capacity-limit of visual working memory is about 3-4 items
However, there is considerable individual variability on capacity limit estimates (k)

Variability between individuals is not just noise

Visual working memory as an example

High and low capacity individuals show systematic differences on behavioral and neural indices of capacity when the set-size exceeds their capacity limitations
This has been interpreted as differences in the ability to selectively filter out irrelevant information
Therefore, capacity limitations may reflect attention processes and not just storage limitations

A latent variable approach to individual differences

Cognitive Psychometric Approach

Individual differences research relies on the use of multiple measures to get at the constructs of interest.
Task impurity: individual task performance will be due to construct of interest, task-specific variance, construct unrelated variance, and error variance.

Experimental approach focuses on manipulations within a single task paradigm
Individual differences approach focuses on extracting common variance across multiple task paradigms

Generally speaking, as psychologists, we are often time not interested in performance on any one single task. We are interested in psychological constructs. Performance on a single task can be due to a variety of different reasons. Some of those reasons are going to be related to the construct that we are actually interested in, attention control, but performance will also be driven by other factors. For instance, some tasks might be more spatial, others more verbal, or requiring some math ability, or tasks might require different sensory domains, visual, or auditory. That is performance on any one task is determined by a multiple sources of variability. But we are actually interested in a particular psychological construct, like attention control, not the individual task itself.

One way to go about studying individual differences in psychological constructs is by using an approach called latent variable modeling. What this does is allows you to separate out variability that is common across a set of tasks, from variability that is specific to an individual task. And that common variability can then be attributed to a latent psychological construct that you think is common across that set of tasks. For instance, in the attention measures, what we see as common across these tasks is some general ability to focus attention on task-relevant information and reducing interference from distraction information. Even though the nature of the distracting information and what has to be attended differs across the tasks.

What this does then, is allows you to study the relationship among a set of latent constructs that you are interested in, rather than just looking at performance at the single task level. Based on this approach, what we have found is that attention control is not only highly predictive of a wide range of cognitive abilities, but that it also explains why cognitive abilities are positively correlated with one another.

A latent variable approach to individual differences

Cognitive Psychometric Approach

Individual differences research relies on the use of multiple measures to get at the constructs of interest.
Task impurity: individual task performance will be due to construct of interest, task-specific variance, construct unrelated variance, and error variance.
Latent constructs: common variance across a set of measures

What are latent variables?

Latent variables are hypothetical constructs that we think our tests measure

Examples of latent variables

Cognitive ability

Examples of latent variables

Psychological Distress

Observed vs. Latent Variables

Observed Variables

In your dataset, the scores on the tests you gave participants
Observed variables are indirect measures of theoretical constructs
Observed variables are also called “indicators” or “manifest variables”

Latent Variablse

Unobserved, hypothetical constructs in their “purest form”
Latent variables are not measured directly, but via observed variables
Latent variables are also known as “latent factors”

Latent Variable Example

Fluid Intelligence

Latent Variable Example

Fluid Intelligence

Latent Variable Example

Fluid Intelligence

Latent Variable Example

Fluid Intelligence

Latent Variable Example

Fluid Intelligence

Pros and cons of latent variable approach

Benefits

Draw conclusions about constructs and not tasks
Test models / theories and assess fit of model to data
Flexible and can handle complex models and data
- Can handle missing data
- Can constrain parameters
- Robust estimation methods for non-normal data
- Mediation, latent growth curve, multi-level, and longitudinal modeling

Pros and cons of latent variable approach

Cons

Need a large sample size (200+)
Need multiple indicators for each construct (3+)
Researcher ‘degrees of freedom’ can be overwhelming
Assessing model fit and mispecification can be complicated

Key considerations for designing an individual differences study

Use multiple tasks to measure a single ability
- At least 3 are recommended but can get away with just 2 in some cases
Avoid difference scores
- e.g., Difference in response times between congruent and incongruent trials in a Stroop task
Consider and assess for reliability
- More trials is typically better but need to balance with administration time
Continuous vs. extreme-group designs
- Don’t split your sample into “high” and “low” groups
Restriction of range
- Recruit from as diverse and representative sample pool as possible
Sample size
- N > 200 is ideal

Practical Hands-On Experience

Variability between individuals is not just noise

Visual working memory as an example

High and low capacity individuals show systematic differences on behavioral and neural indices of capacity when the set-size exceeds their capacity limitations
This has been interpreted as differences in the ability to selectively filter out irrelevant information
Therefore, capacity limitations may reflect attention processes and not just storage limitations

Example data set

Martin et al. (2021)

Data Set 3 (N = 568)

Performance on change-detection visual arrays tasks are interpreted as reflecting the limited-capacity storage and maintenance of visual information in working memory. However, there is evidence to suggest that visual arrays task performance reflects individual differences in attention control processes. The purpose of this study was to evaluate the extent to which selective and non-selective visual arrays tasks reflect working memory storage and maintenance vs. attention control processes.

Working memory capacity: Operation span, Symmetry Span, Rotation Span, Running Spatial Span, Running Digit Span

Attention control: Antisaccade, Flanker, Stroop

Example data set

Martin et al. (2021)

Data Set 3 (N = 568)

Analysis steps

Data quality check
- Assess amount of missing data (<5%), outliers, distributions (skewness and kurtosis)
Correlation matrix
- Assess and familiarize yourself with the correlation among variables
Exporatory factor analysis
- Data-driven approach to identifying latent factors
Confirmatory factor analysis
- Theory-driven approach to test how well our model/factor structure fits the data
- Assess the correlations between latent factors
Structural equation modeling
- Regression-based approach to model relations between latent factors

Analysis steps

Data quality check
- Assess amount of missing data (<5%), outliers, distributions (skewness and kurtosis)
Correlation matrix
- Assess and familiarize yourself with the correlation among variables
Exporatory factor analysis
- Data-driven approach to identifying latent factors
Confirmatory factor analysis
- Theory-driven approach to test how well our model/factor structure fits the data
- Assess the correlations between latent factors
Structural equation modeling
- Regression-based approach to model relations between latent factors

Quarto template for analysis

Data quality check

library(semoutput)

sem_descriptives(data)

Variable	n	Mean	SD	Range	Skewness	Kurtosis	% Missing
Descriptive Statistics
OSpan	564	54.440	15.338	3.000 — 75.000	−0.878	0.145	0.704
SymSpan	565	26.644	9.056	3.000 — 42.000	−0.417	−0.502	0.528
RotSpan	565	24.664	9.805	0.000 — 42.000	−0.448	−0.577	0.528
RunSpatial	567	24.577	11.288	1.000 — 49.000	−0.310	−0.798	0.176
RunDigit	561	56.624	19.360	0.000 — 107.000	−0.458	0.225	1.232
Antisaccade	566	0.556	0.163	0.200 — 0.983	0.116	−0.793	0.352
Flanker	561	98.460	44.322	−49.966 — 251.245	0.629	1.081	1.232
Stroop	567	131.552	98.160	−152.632 — 476.290	0.670	0.933	0.176
VAorient_k	568	2.747	1.492	−2.467 — 5.767	−0.490	−0.126	0.000
VAorient_S_k	568	1.586	1.298	−1.675 — 5.000	0.101	−0.471	0.000
Total N = 568

Standard guidelines:

< 5% of missing data on any single variable
Outliers: +/- 3 or 3.5 SD
Skewness: within +/- 3
Kurtosis: within +/- 10

Correlations

library(sjPlot)

tab_corr(data, na.deletion = "pairwise", show.p = FALSE)

	OSpan	SymSpan	RotSpan	RunSpatial	RunDigit	Antisaccade	Flanker	Stroop	VAorient_k	VAorient_S_k
OSpan		0.542	0.528	0.529	0.521	0.378	-0.114	-0.303	0.363	0.432
SymSpan	0.542		0.683	0.615	0.435	0.446	-0.148	-0.185	0.462	0.512
RotSpan	0.528	0.683		0.603	0.489	0.440	-0.165	-0.180	0.432	0.504
RunSpatial	0.529	0.615	0.603		0.558	0.480	-0.223	-0.240	0.553	0.584
RunDigit	0.521	0.435	0.489	0.558		0.405	-0.144	-0.233	0.352	0.471
Antisaccade	0.378	0.446	0.440	0.480	0.405		-0.194	-0.221	0.383	0.497
Flanker	-0.114	-0.148	-0.165	-0.223	-0.144	-0.194		0.093	-0.183	-0.160
Stroop	-0.303	-0.185	-0.180	-0.240	-0.233	-0.221	0.093		-0.170	-0.206
VAorient_k	0.363	0.462	0.432	0.553	0.352	0.383	-0.183	-0.170		0.577
VAorient_S_k	0.432	0.512	0.504	0.584	0.471	0.497	-0.160	-0.206	0.577
Computed correlation used pearson-method with pairwise-deletion.

What to look for:

Convergent validity: Stronger correlations between tasks within a construct
Discriminate validity: Weaker correlations between tasks across constructs
Multicolinearity: Too high of correlations, r > .85
Weak correlations

Latent variable analyses, model the variance covariance matrix

	OSpan	SymSpan	RotSpan	RunSpatial	RunDigit	Antisaccade	Flanker	Stroop	VAorient_k	VAorient_S_k
OSpan	235.263	75.042	78.716	91.570	154.288	0.940	−77.752	−453.431	8.278	8.608
SymSpan	75.042	82.010	60.840	62.972	76.208	0.658	−59.711	−164.515	6.248	6.034
RotSpan	78.716	60.840	96.142	66.639	92.675	0.703	−71.974	−173.089	6.308	6.398
RunSpatial	91.570	62.972	66.639	127.428	121.168	0.883	−111.187	−265.489	9.318	8.560
RunDigit	154.288	76.208	92.675	121.168	374.828	1.268	−123.684	−442.762	10.205	11.786
Antisaccade	0.940	0.658	0.703	0.883	1.268	0.027	−1.399	−3.544	0.093	0.105
Flanker	−77.752	−59.711	−71.974	−111.187	−123.684	−1.399	1,964.462	406.326	−11.987	−9.202
Stroop	−453.431	−164.515	−173.089	−265.489	−442.762	−3.544	406.326	9,635.449	−24.909	−26.159
VAorient_k	8.278	6.248	6.308	9.318	10.205	0.093	−11.987	−24.909	2.225	1.116
VAorient_S_k	8.608	6.034	6.398	8.560	11.786	0.105	−9.202	−26.159	1.116	1.684

∑_observed = ∑_model + ∑_residual

Model fit is assessed by how closely our model reproduces (or explains) the observed variance covariance matrix
You need to balance parsimony (a simpler model with fewer parameter estimates) and adequate model fit (a more complex model with theoretically irrelevant parameters)

Confirmatory factor analysis: Specify model syntax

`lavaan::cfa()`

	OSpan	SymSpan	RotSpan	RunSpatial	RunDigit	Antisaccade	Flanker	Stroop	VAorient_k	VAorient_S_k
OSpan	235.263	75.042	78.716	91.570	154.288	0.940	−77.752	−453.431	8.278	8.608
SymSpan	75.042	82.010	60.840	62.972	76.208	0.658	−59.711	−164.515	6.248	6.034
RotSpan	78.716	60.840	96.142	66.639	92.675	0.703	−71.974	−173.089	6.308	6.398
RunSpatial	91.570	62.972	66.639	127.428	121.168	0.883	−111.187	−265.489	9.318	8.560
RunDigit	154.288	76.208	92.675	121.168	374.828	1.268	−123.684	−442.762	10.205	11.786
Antisaccade	0.940	0.658	0.703	0.883	1.268	0.027	−1.399	−3.544	0.093	0.105
Flanker	−77.752	−59.711	−71.974	−111.187	−123.684	−1.399	1,964.462	406.326	−11.987	−9.202
Stroop	−453.431	−164.515	−173.089	−265.489	−442.762	−3.544	406.326	9,635.449	−24.909	−26.159
VAorient_k	8.278	6.248	6.308	9.318	10.205	0.093	−11.987	−24.909	2.225	1.116
VAorient_S_k	8.608	6.034	6.398	8.560	11.786	0.105	−9.202	−26.159	1.116	1.684

∑_observed = ∑_model + ∑_residual

Model fit is assessed by how closely our model reproduces (or explains) the observed variance covariance matrix
You need to balance parsimony (a simpler model with fewer parameter estimates) and adequate model fit (a more complex model with theoretically irrelevant parameters)

Confirmatory factor analysis: Specify model syntax

`lavaan::cfa()`

model <- '
# latent factors
WMC =~ OSpan + SymSpan + RotSpan + RunSpatial + RunDigit
AC =~ Antisaccade + Flanker + Stroop

# single-indicator factors
VA_ns =~ 1*VAorient_k
VA_s =~ 1*VAorient_S_k
'

The model syntax contains formula types enclosed in single brackets
- Formula type to define latent factor: f1 =~ ind1 + ind2 + ind3

Confirmatory factor analysis: Fit the model

`lavaan::cfa()`

model <- '
# latent factors
WMC =~ OSpan + SymSpan + RotSpan + RunSpatial + RunDigit
AC =~ Antisaccade + Flanker + Stroop

# single-indicator factors
VA_ns =~ 1*VAorient_k
VA_s =~ 1*VAorient_S_k
'

fit <- cfa(model, data, missing = "fiml", std.ov = TRUE)

Fit the model with cfa()
- Specify the model syntax object, the data, and additional arguments
- missing = "fiml" — uses full information maximum likelihood. This is the best method for dealing with missing data
- std.ov = TRUE — standardizes the observed (indicator) variables before running the model (i.e., uses a correlation matrix instead of covariance matrix)

Confirmatory factor analysis: Assess model output

`lavaan::cfa()`

summary(fit, fit.measures = TRUE, standardized = TRUE)

lavaan 0.6-19 ended normally after 34 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                        34

  Number of observations                           568
  Number of missing patterns                        10

Model Test User Model:
                                                      
  Test statistic                               107.514
  Degrees of freedom                                31
  P-value (Chi-square)                           0.000

Model Test Baseline Model:

  Test statistic                              2163.395
  Degrees of freedom                                45
  P-value                                        0.000

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.964
  Tucker-Lewis Index (TLI)                       0.948
                                                      
  Robust Comparative Fit Index (CFI)             0.964
  Robust Tucker-Lewis Index (TLI)                0.947

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -6986.895
  Loglikelihood unrestricted model (H1)      -6933.138
                                                      
  Akaike (AIC)                               14041.791
  Bayesian (BIC)                             14189.423
  Sample-size adjusted Bayesian (SABIC)      14081.489

Root Mean Square Error of Approximation:

  RMSEA                                          0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: RMSEA <= 0.050                    0.026
  P-value H_0: RMSEA >= 0.080                    0.047
                                                      
  Robust RMSEA                                   0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: Robust RMSEA <= 0.050             0.023
  P-value H_0: Robust RMSEA >= 0.080             0.055

Standardized Root Mean Square Residual:

  SRMR                                           0.030

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Observed
  Observed information based on                Hessian

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC =~                                                                
    OSpan             1.000                               0.682    0.682
    SymSpan           1.142    0.070   16.321    0.000    0.779    0.780
    RotSpan           1.141    0.070   16.270    0.000    0.778    0.779
    RunSpatial        1.185    0.071   16.675    0.000    0.809    0.810
    RunDigit          0.963    0.068   14.104    0.000    0.657    0.656
  AC =~                                                                 
    Antisaccade       1.000                               0.698    0.699
    Flanker          -0.377    0.071   -5.336    0.000   -0.263   -0.264
    Stroop           -0.465    0.073   -6.417    0.000   -0.325   -0.325
  VA_ns =~                                                              
    VAorient_k        1.000                               0.999    1.000
  VA_s =~                                                               
    VAorient_S_k      1.000                               0.999    1.000

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC ~~                                                                
    AC                0.399    0.039   10.167    0.000    0.838    0.838
    VA_ns             0.406    0.040   10.270    0.000    0.596    0.596
    VA_s              0.462    0.042   11.032    0.000    0.677    0.677
  AC ~~                                                                 
    VA_ns             0.388    0.044    8.818    0.000    0.556    0.556
    VA_s              0.486    0.047   10.441    0.000    0.696    0.696
  VA_ns ~~                                                              
    VA_s              0.575    0.048   11.903    0.000    0.577    0.577

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan            -0.006    0.042   -0.134    0.894   -0.006   -0.006
   .SymSpan          -0.001    0.042   -0.036    0.972   -0.001   -0.001
   .RotSpan           0.001    0.042    0.017    0.986    0.001    0.001
   .RunSpatial       -0.000    0.042   -0.009    0.993   -0.000   -0.000
   .RunDigit          0.006    0.042    0.142    0.887    0.006    0.006
   .Antisaccade      -0.001    0.042   -0.025    0.980   -0.001   -0.001
   .Flanker           0.002    0.042    0.053    0.958    0.002    0.002
   .Stroop            0.001    0.042    0.013    0.990    0.001    0.001
   .VAorient_k       -0.000    0.042   -0.000    1.000   -0.000   -0.000
   .VAorient_S_k     -0.000    0.042   -0.000    1.000   -0.000   -0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan             0.536    0.036   14.906    0.000    0.536    0.535
   .SymSpan           0.390    0.029   13.261    0.000    0.390    0.391
   .RotSpan           0.393    0.030   13.295    0.000    0.393    0.394
   .RunSpatial        0.343    0.027   12.559    0.000    0.343    0.344
   .RunDigit          0.570    0.038   15.056    0.000    0.570    0.569
   .Antisaccade       0.511    0.067    7.674    0.000    0.511    0.512
   .Flanker           0.929    0.057   16.313    0.000    0.929    0.931
   .Stroop            0.893    0.055   16.178    0.000    0.893    0.894
   .VAorient_k        0.000                               0.000    0.000
   .VAorient_S_k      0.000                               0.000    0.000
    WMC               0.465    0.053    8.764    0.000    1.000    1.000
    AC                0.488    0.078    6.248    0.000    1.000    1.000
    VA_ns             0.998    0.059   16.852    0.000    1.000    1.000
    VA_s              0.998    0.059   16.852    0.000    1.000    1.000

Print model output with summary()
- fit.measures = TRUE — Outputs measures to assess the fit of the model
- standardized = TRUE — Outputs standardized model parameters

Confirmatory factor analysis: Assess model output

`lavaan::cfa()`

summary(fit, fit.measures = TRUE, standardized = TRUE)

lavaan 0.6-19 ended normally after 34 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                        34

  Number of observations                           568
  Number of missing patterns                        10

Model Test User Model:
                                                      
  Test statistic                               107.514
  Degrees of freedom                                31
  P-value (Chi-square)                           0.000

Model Test Baseline Model:

  Test statistic                              2163.395
  Degrees of freedom                                45
  P-value                                        0.000

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.964
  Tucker-Lewis Index (TLI)                       0.948
                                                      
  Robust Comparative Fit Index (CFI)             0.964
  Robust Tucker-Lewis Index (TLI)                0.947

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -6986.895
  Loglikelihood unrestricted model (H1)      -6933.138
                                                      
  Akaike (AIC)                               14041.791
  Bayesian (BIC)                             14189.423
  Sample-size adjusted Bayesian (SABIC)      14081.489

Root Mean Square Error of Approximation:

  RMSEA                                          0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: RMSEA <= 0.050                    0.026
  P-value H_0: RMSEA >= 0.080                    0.047
                                                      
  Robust RMSEA                                   0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: Robust RMSEA <= 0.050             0.023
  P-value H_0: Robust RMSEA >= 0.080             0.055

Standardized Root Mean Square Residual:

  SRMR                                           0.030

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Observed
  Observed information based on                Hessian

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC =~                                                                
    OSpan             1.000                               0.682    0.682
    SymSpan           1.142    0.070   16.321    0.000    0.779    0.780
    RotSpan           1.141    0.070   16.270    0.000    0.778    0.779
    RunSpatial        1.185    0.071   16.675    0.000    0.809    0.810
    RunDigit          0.963    0.068   14.104    0.000    0.657    0.656
  AC =~                                                                 
    Antisaccade       1.000                               0.698    0.699
    Flanker          -0.377    0.071   -5.336    0.000   -0.263   -0.264
    Stroop           -0.465    0.073   -6.417    0.000   -0.325   -0.325
  VA_ns =~                                                              
    VAorient_k        1.000                               0.999    1.000
  VA_s =~                                                               
    VAorient_S_k      1.000                               0.999    1.000

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC ~~                                                                
    AC                0.399    0.039   10.167    0.000    0.838    0.838
    VA_ns             0.406    0.040   10.270    0.000    0.596    0.596
    VA_s              0.462    0.042   11.032    0.000    0.677    0.677
  AC ~~                                                                 
    VA_ns             0.388    0.044    8.818    0.000    0.556    0.556
    VA_s              0.486    0.047   10.441    0.000    0.696    0.696
  VA_ns ~~                                                              
    VA_s              0.575    0.048   11.903    0.000    0.577    0.577

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan            -0.006    0.042   -0.134    0.894   -0.006   -0.006
   .SymSpan          -0.001    0.042   -0.036    0.972   -0.001   -0.001
   .RotSpan           0.001    0.042    0.017    0.986    0.001    0.001
   .RunSpatial       -0.000    0.042   -0.009    0.993   -0.000   -0.000
   .RunDigit          0.006    0.042    0.142    0.887    0.006    0.006
   .Antisaccade      -0.001    0.042   -0.025    0.980   -0.001   -0.001
   .Flanker           0.002    0.042    0.053    0.958    0.002    0.002
   .Stroop            0.001    0.042    0.013    0.990    0.001    0.001
   .VAorient_k       -0.000    0.042   -0.000    1.000   -0.000   -0.000
   .VAorient_S_k     -0.000    0.042   -0.000    1.000   -0.000   -0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan             0.536    0.036   14.906    0.000    0.536    0.535
   .SymSpan           0.390    0.029   13.261    0.000    0.390    0.391
   .RotSpan           0.393    0.030   13.295    0.000    0.393    0.394
   .RunSpatial        0.343    0.027   12.559    0.000    0.343    0.344
   .RunDigit          0.570    0.038   15.056    0.000    0.570    0.569
   .Antisaccade       0.511    0.067    7.674    0.000    0.511    0.512
   .Flanker           0.929    0.057   16.313    0.000    0.929    0.931
   .Stroop            0.893    0.055   16.178    0.000    0.893    0.894
   .VAorient_k        0.000                               0.000    0.000
   .VAorient_S_k      0.000                               0.000    0.000
    WMC               0.465    0.053    8.764    0.000    1.000    1.000
    AC                0.488    0.078    6.248    0.000    1.000    1.000
    VA_ns             0.998    0.059   16.852    0.000    1.000    1.000
    VA_s              0.998    0.059   16.852    0.000    1.000    1.000

Confirmatory factor analysis: Assess model output

`lavaan::cfa()`

residuals(fit, type = "cor")$cov

              OSpan SymSpn RotSpn RnSptl RunDgt Antscc Flankr Stroop VArnt_
OSpan         0.000                                                        
SymSpan       0.008  0.000                                                 
RotSpan       0.000  0.075  0.000                                          
RunSpatial   -0.022 -0.018 -0.029  0.000                                   
RunDigit      0.076 -0.074 -0.020  0.025  0.000                            
Antisaccade  -0.021 -0.011 -0.015  0.007  0.026  0.000                     
Flanker       0.040  0.024  0.006 -0.044 -0.004 -0.009  0.000              
Stroop       -0.117  0.029  0.032 -0.018 -0.054  0.009  0.010  0.000       
VAorient_k   -0.043 -0.002 -0.033  0.070 -0.037 -0.005 -0.040  0.011  0.000
VAorient_S_k -0.029 -0.017 -0.023  0.035  0.028  0.010  0.023  0.022  0.000
             VAr_S_
OSpan              
SymSpan            
RotSpan            
RunSpatial         
RunDigit           
Antisaccade        
Flanker            
Stroop             
VAorient_k         
VAorient_S_k  0.000

∑_observed = ∑_model + ∑_residual

Model fit is assessed by how closely our model reproduces (or explains) the observed variance covariance matrix
You need to balance parsimony (a simpler model with fewer parameter estimates) and adequate model fit (a more complex model with theoretically irrelevant parameters)

Confirmatory factor analysis: Assess model output

`lavaan::cfa()`

summary(fit, fit.measures = TRUE, standardized = TRUE)

lavaan 0.6-19 ended normally after 34 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                        34

  Number of observations                           568
  Number of missing patterns                        10

Model Test User Model:
                                                      
  Test statistic                               107.514
  Degrees of freedom                                31
  P-value (Chi-square)                           0.000

Model Test Baseline Model:

  Test statistic                              2163.395
  Degrees of freedom                                45
  P-value                                        0.000

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.964
  Tucker-Lewis Index (TLI)                       0.948
                                                      
  Robust Comparative Fit Index (CFI)             0.964
  Robust Tucker-Lewis Index (TLI)                0.947

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -6986.895
  Loglikelihood unrestricted model (H1)      -6933.138
                                                      
  Akaike (AIC)                               14041.791
  Bayesian (BIC)                             14189.423
  Sample-size adjusted Bayesian (SABIC)      14081.489

Root Mean Square Error of Approximation:

  RMSEA                                          0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: RMSEA <= 0.050                    0.026
  P-value H_0: RMSEA >= 0.080                    0.047
                                                      
  Robust RMSEA                                   0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: Robust RMSEA <= 0.050             0.023
  P-value H_0: Robust RMSEA >= 0.080             0.055

Standardized Root Mean Square Residual:

  SRMR                                           0.030

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Observed
  Observed information based on                Hessian

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC =~                                                                
    OSpan             1.000                               0.682    0.682
    SymSpan           1.142    0.070   16.321    0.000    0.779    0.780
    RotSpan           1.141    0.070   16.270    0.000    0.778    0.779
    RunSpatial        1.185    0.071   16.675    0.000    0.809    0.810
    RunDigit          0.963    0.068   14.104    0.000    0.657    0.656
  AC =~                                                                 
    Antisaccade       1.000                               0.698    0.699
    Flanker          -0.377    0.071   -5.336    0.000   -0.263   -0.264
    Stroop           -0.465    0.073   -6.417    0.000   -0.325   -0.325
  VA_ns =~                                                              
    VAorient_k        1.000                               0.999    1.000
  VA_s =~                                                               
    VAorient_S_k      1.000                               0.999    1.000

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC ~~                                                                
    AC                0.399    0.039   10.167    0.000    0.838    0.838
    VA_ns             0.406    0.040   10.270    0.000    0.596    0.596
    VA_s              0.462    0.042   11.032    0.000    0.677    0.677
  AC ~~                                                                 
    VA_ns             0.388    0.044    8.818    0.000    0.556    0.556
    VA_s              0.486    0.047   10.441    0.000    0.696    0.696
  VA_ns ~~                                                              
    VA_s              0.575    0.048   11.903    0.000    0.577    0.577

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan            -0.006    0.042   -0.134    0.894   -0.006   -0.006
   .SymSpan          -0.001    0.042   -0.036    0.972   -0.001   -0.001
   .RotSpan           0.001    0.042    0.017    0.986    0.001    0.001
   .RunSpatial       -0.000    0.042   -0.009    0.993   -0.000   -0.000
   .RunDigit          0.006    0.042    0.142    0.887    0.006    0.006
   .Antisaccade      -0.001    0.042   -0.025    0.980   -0.001   -0.001
   .Flanker           0.002    0.042    0.053    0.958    0.002    0.002
   .Stroop            0.001    0.042    0.013    0.990    0.001    0.001
   .VAorient_k       -0.000    0.042   -0.000    1.000   -0.000   -0.000
   .VAorient_S_k     -0.000    0.042   -0.000    1.000   -0.000   -0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan             0.536    0.036   14.906    0.000    0.536    0.535
   .SymSpan           0.390    0.029   13.261    0.000    0.390    0.391
   .RotSpan           0.393    0.030   13.295    0.000    0.393    0.394
   .RunSpatial        0.343    0.027   12.559    0.000    0.343    0.344
   .RunDigit          0.570    0.038   15.056    0.000    0.570    0.569
   .Antisaccade       0.511    0.067    7.674    0.000    0.511    0.512
   .Flanker           0.929    0.057   16.313    0.000    0.929    0.931
   .Stroop            0.893    0.055   16.178    0.000    0.893    0.894
   .VAorient_k        0.000                               0.000    0.000
   .VAorient_S_k      0.000                               0.000    0.000
    WMC               0.465    0.053    8.764    0.000    1.000    1.000
    AC                0.488    0.078    6.248    0.000    1.000    1.000
    VA_ns             0.998    0.059   16.852    0.000    1.000    1.000
    VA_s              0.998    0.059   16.852    0.000    1.000    1.000

Standardized loading can be found in Std.all column and should be:
- Statistically significant, p < .05
- Medium to large in size (standardized loading), > .30
- Fairly balanced across the factor

Confirmatory factor analysis: Assess model output

`lavaan::cfa()`

summary(fit, fit.measures = TRUE, standardized = TRUE)

lavaan 0.6-19 ended normally after 34 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                        34

  Number of observations                           568
  Number of missing patterns                        10

Model Test User Model:
                                                      
  Test statistic                               107.514
  Degrees of freedom                                31
  P-value (Chi-square)                           0.000

Model Test Baseline Model:

  Test statistic                              2163.395
  Degrees of freedom                                45
  P-value                                        0.000

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.964
  Tucker-Lewis Index (TLI)                       0.948
                                                      
  Robust Comparative Fit Index (CFI)             0.964
  Robust Tucker-Lewis Index (TLI)                0.947

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -6986.895
  Loglikelihood unrestricted model (H1)      -6933.138
                                                      
  Akaike (AIC)                               14041.791
  Bayesian (BIC)                             14189.423
  Sample-size adjusted Bayesian (SABIC)      14081.489

Root Mean Square Error of Approximation:

  RMSEA                                          0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: RMSEA <= 0.050                    0.026
  P-value H_0: RMSEA >= 0.080                    0.047
                                                      
  Robust RMSEA                                   0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: Robust RMSEA <= 0.050             0.023
  P-value H_0: Robust RMSEA >= 0.080             0.055

Standardized Root Mean Square Residual:

  SRMR                                           0.030

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Observed
  Observed information based on                Hessian

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC =~                                                                
    OSpan             1.000                               0.682    0.682
    SymSpan           1.142    0.070   16.321    0.000    0.779    0.780
    RotSpan           1.141    0.070   16.270    0.000    0.778    0.779
    RunSpatial        1.185    0.071   16.675    0.000    0.809    0.810
    RunDigit          0.963    0.068   14.104    0.000    0.657    0.656
  AC =~                                                                 
    Antisaccade       1.000                               0.698    0.699
    Flanker          -0.377    0.071   -5.336    0.000   -0.263   -0.264
    Stroop           -0.465    0.073   -6.417    0.000   -0.325   -0.325
  VA_ns =~                                                              
    VAorient_k        1.000                               0.999    1.000
  VA_s =~                                                               
    VAorient_S_k      1.000                               0.999    1.000

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC ~~                                                                
    AC                0.399    0.039   10.167    0.000    0.838    0.838
    VA_ns             0.406    0.040   10.270    0.000    0.596    0.596
    VA_s              0.462    0.042   11.032    0.000    0.677    0.677
  AC ~~                                                                 
    VA_ns             0.388    0.044    8.818    0.000    0.556    0.556
    VA_s              0.486    0.047   10.441    0.000    0.696    0.696
  VA_ns ~~                                                              
    VA_s              0.575    0.048   11.903    0.000    0.577    0.577

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan            -0.006    0.042   -0.134    0.894   -0.006   -0.006
   .SymSpan          -0.001    0.042   -0.036    0.972   -0.001   -0.001
   .RotSpan           0.001    0.042    0.017    0.986    0.001    0.001
   .RunSpatial       -0.000    0.042   -0.009    0.993   -0.000   -0.000
   .RunDigit          0.006    0.042    0.142    0.887    0.006    0.006
   .Antisaccade      -0.001    0.042   -0.025    0.980   -0.001   -0.001
   .Flanker           0.002    0.042    0.053    0.958    0.002    0.002
   .Stroop            0.001    0.042    0.013    0.990    0.001    0.001
   .VAorient_k       -0.000    0.042   -0.000    1.000   -0.000   -0.000
   .VAorient_S_k     -0.000    0.042   -0.000    1.000   -0.000   -0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan             0.536    0.036   14.906    0.000    0.536    0.535
   .SymSpan           0.390    0.029   13.261    0.000    0.390    0.391
   .RotSpan           0.393    0.030   13.295    0.000    0.393    0.394
   .RunSpatial        0.343    0.027   12.559    0.000    0.343    0.344
   .RunDigit          0.570    0.038   15.056    0.000    0.570    0.569
   .Antisaccade       0.511    0.067    7.674    0.000    0.511    0.512
   .Flanker           0.929    0.057   16.313    0.000    0.929    0.931
   .Stroop            0.893    0.055   16.178    0.000    0.893    0.894
   .VAorient_k        0.000                               0.000    0.000
   .VAorient_S_k      0.000                               0.000    0.000
    WMC               0.465    0.053    8.764    0.000    1.000    1.000
    AC                0.488    0.078    6.248    0.000    1.000    1.000
    VA_ns             0.998    0.059   16.852    0.000    1.000    1.000
    VA_s              0.998    0.059   16.852    0.000    1.000    1.000

Correlations between latent factors can be found in Covariances section
- WMC and AC are correlated about the same with non-selective and selective visual arrays

Confirmatory factor analysis: Assess model output

`lavaan::cfa()`

Alternatively, print model output with semoutput::sem_tables(fit)

sem_tables(fit)

N	χ²	df	p
Model Significance
568	107.514	31	<0.001

CFI	RMSEA	90% CI	TLI	SRMR	AIC	BIC
Model Fit
0.964	0.066	0.053 — 0.080	0.948	0.030	14041.791	14189.423

Latent Factor	Indicator	Standardized
Factor Loadings
Latent Factor	Indicator	Loading	95% CI	sig	SE	z	p
AC	Antisaccade	0.699	0.606 — 0.792	***	0.048	14.699	0.000
AC	Flanker	−0.264	−0.354 — −0.173	***	0.046	−5.681	<0.001
AC	Stroop	−0.325	−0.412 — −0.238	***	0.044	−7.339	<0.001
VA_ns	VAorient_k	1.000	1.000 — 1.000		0.000
VA_s	VAorient_S_k	1.000	1.000 — 1.000		0.000
WMC	OSpan	0.682	0.632 — 0.732	***	0.026	26.608	0.000
WMC	RotSpan	0.779	0.739 — 0.818	***	0.020	38.477	0.000
WMC	RunDigit	0.656	0.603 — 0.710	***	0.027	24.136	0.000
WMC	RunSpatial	0.810	0.774 — 0.846	***	0.018	44.163	0.000
WMC	SymSpan	0.780	0.741 — 0.820	***	0.020	38.781	0.000
* p < .05; p < .01; * p < .001

Factor	Factor	r	95% CI	sig	SE	p
Latent Factor Correlations
WMC	AC	0.838	0.728 — 0.948	***	0.056	0.000
WMC	VA_ns	0.596	0.536 — 0.655	***	0.030	0.000
WMC	VA_s	0.677	0.626 — 0.728	***	0.026	0.000
AC	VA_ns	0.556	0.451 — 0.661	***	0.054	0.000
AC	VA_s	0.696	0.597 — 0.795	***	0.051	0.000
VA_ns	VA_s	0.577	0.522 — 0.631	***	0.028	0.000
* p < .05; p < .01; * p < .001

Factor	Variance	Std. Variance	sig	p
Latent Factor Variance/Residual Variance
WMC	0.465	1.000	***	0.000
AC	0.488	1.000	***	<0.001
VA_ns	0.998	1.000	***	0.000
VA_s	0.998	1.000	***	0.000

From research question to structural model

Martin et al. (2021)

Data Set 3 (N = 568)

The purpose of this study was to evaluate the extent to which selective and non-selective visual arrays tasks reflect working memory storage and maintenance (WMC) vs. attention control processes (AC).

From research question to structural model

Martin et al. (2021)

Data Set 3 (N = 568)

What is the unique contribution of working memory capacity and attention control to variability in non-selective and selective visual arrays?

Differences from a CFA:

Similar to correlation vs. multiple regression
Partitions unique sources of variance
Directional paths (⟹) instead of bi-directional paths (⟷)
Residual variance terms for dependent variables

What is the unique contribution of WMC and AC to variability in visual arrays tasks?

Structural equation model: Fit the model

model <- '
# latent factors
WMC =~ NA*OSpan + SymSpan + RotSpan + RunSpatial + RunDigit
AC =~ NA*Antisaccade + Flanker + Stroop

WMC ~~ 1*WMC
AC ~~ 1*AC

# single-indicator factors
VA_ns =~ 1*VAorient_k
VA_s =~ 1*VAorient_S_k

# regression paths
VA_ns ~ WMC + AC
VA_s ~ WMC + AC
'

fit <- sem(model, data, missing = "fiml", std.ov = TRUE)

It is recommended to set:

Exogenous (predictor) latent factor variances to 1
- Set first factor loading to be freely estimated: NA*
- Set latent factor variance to 1: WMC ~~ 1*WMC
Endogenous (dependent variable) latent factor variance to be freely estimated

What is the unique contribution of WMC and AC to variability in visual arrays tasks?

Structural equation model: Fit the model

model <- '
# latent factors
WMC =~ NA*OSpan + SymSpan + RotSpan + RunSpatial + RunDigit
AC =~ NA*Antisaccade + Flanker + Stroop

WMC ~~ 1*WMC
AC ~~ 1*AC

# single-indicator factors
VA_ns =~ 1*VAorient_k
VA_s =~ 1*VAorient_S_k

# regression paths
VA_ns ~ WMC + AC
VA_s ~ WMC + AC
'

fit <- sem(model, data, missing = "fiml", std.ov = TRUE)

Model syntax formula types

NA* Freely estimate loading — default in sem() is to set first factor loading to 1
~~ Set a variance or co-variance component
~ Regression formula

What is the unique contribution of WMC and AC to variability in visual arrays tasks?

Structural equation model: Assess the model

summary(fit, fit.measures = TRUE, standardized = TRUE)

lavaan 0.6-19 ended normally after 61 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                        34

  Number of observations                           568
  Number of missing patterns                        10

Model Test User Model:
                                                      
  Test statistic                               107.514
  Degrees of freedom                                31
  P-value (Chi-square)                           0.000

Model Test Baseline Model:

  Test statistic                              2163.395
  Degrees of freedom                                45
  P-value                                        0.000

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.964
  Tucker-Lewis Index (TLI)                       0.948
                                                      
  Robust Comparative Fit Index (CFI)             0.964
  Robust Tucker-Lewis Index (TLI)                0.947

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -6986.895
  Loglikelihood unrestricted model (H1)      -6933.138
                                                      
  Akaike (AIC)                               14041.791
  Bayesian (BIC)                             14189.423
  Sample-size adjusted Bayesian (SABIC)      14081.489

Root Mean Square Error of Approximation:

  RMSEA                                          0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: RMSEA <= 0.050                    0.026
  P-value H_0: RMSEA >= 0.080                    0.047
                                                      
  Robust RMSEA                                   0.066
  90 Percent confidence interval - lower         0.053
  90 Percent confidence interval - upper         0.080
  P-value H_0: Robust RMSEA <= 0.050             0.023
  P-value H_0: Robust RMSEA >= 0.080             0.055

Standardized Root Mean Square Residual:

  SRMR                                           0.030

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Observed
  Observed information based on                Hessian

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC =~                                                                
    OSpan             0.682    0.039   17.527    0.000    0.682    0.682
    SymSpan           0.779    0.037   21.149    0.000    0.779    0.780
    RotSpan           0.778    0.037   21.059    0.000    0.778    0.779
    RunSpatial        0.809    0.036   22.365    0.000    0.809    0.810
    RunDigit          0.657    0.040   16.596    0.000    0.657    0.656
  AC =~                                                                 
    Antisaccade       0.698    0.056   12.495    0.000    0.698    0.699
    Flanker          -0.263    0.048   -5.460    0.000   -0.263   -0.264
    Stroop           -0.325    0.047   -6.897    0.000   -0.325   -0.325
  VA_ns =~                                                              
    VAorient_k        1.000                               0.999    1.000
  VA_s =~                                                               
    VAorient_S_k      1.000                               0.999    1.000

Regressions:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  VA_ns ~                                                               
    WMC               0.436    0.133    3.272    0.001    0.436    0.436
    AC                0.190    0.143    1.324    0.185    0.190    0.190
  VA_s ~                                                                
    WMC               0.315    0.167    1.891    0.059    0.315    0.315
    AC                0.431    0.173    2.500    0.012    0.432    0.432

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  WMC ~~                                                                
    AC                0.838    0.056   14.972    0.000    0.838    0.838
 .VA_ns ~~                                                              
   .VA_s              0.148    0.031    4.764    0.000    0.268    0.268

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .OSpan            -0.006    0.042   -0.134    0.894   -0.006   -0.006
   .SymSpan          -0.001    0.042   -0.036    0.972   -0.001   -0.001
   .RotSpan           0.001    0.042    0.017    0.986    0.001    0.001
   .RunSpatial       -0.000    0.042   -0.009    0.993   -0.000   -0.000
   .RunDigit          0.006    0.042    0.142    0.887    0.006    0.006
   .Antisaccade      -0.001    0.042   -0.025    0.980   -0.001   -0.001
   .Flanker           0.002    0.042    0.053    0.958    0.002    0.002
   .Stroop            0.001    0.042    0.013    0.990    0.001    0.001
   .VAorient_k        0.000    0.042    0.000    1.000    0.000    0.000
   .VAorient_S_k     -0.000    0.042   -0.000    1.000   -0.000   -0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
    WMC               1.000                               1.000    1.000
    AC                1.000                               1.000    1.000
   .OSpan             0.536    0.036   14.906    0.000    0.536    0.535
   .SymSpan           0.390    0.029   13.261    0.000    0.390    0.391
   .RotSpan           0.393    0.030   13.295    0.000    0.393    0.394
   .RunSpatial        0.343    0.027   12.559    0.000    0.343    0.344
   .RunDigit          0.570    0.038   15.056    0.000    0.570    0.569
   .Antisaccade       0.511    0.067    7.674    0.000    0.511    0.512
   .Flanker           0.929    0.057   16.313    0.000    0.929    0.931
   .Stroop            0.893    0.055   16.178    0.000    0.893    0.894
   .VAorient_k        0.000                               0.000    0.000
   .VAorient_S_k      0.000                               0.000    0.000
   .VA_ns             0.633    0.041   15.295    0.000    0.635    0.635
   .VA_s              0.485    0.042   11.443    0.000    0.486    0.486

Path values can be found in the Regression section

What is the unique contribution of WMC and AC to variability in visual arrays tasks?

Structural equation model: Assess the model

sem_tables(fit)

N	χ²	df	p
Model Significance
568	107.514	31	<0.001

CFI	RMSEA	90% CI	TLI	SRMR	AIC	BIC
Model Fit
0.964	0.066	0.053 — 0.080	0.948	0.030	14041.791	14189.423

Latent Factor	Indicator	Standardized
Factor Loadings
Latent Factor	Indicator	Loading	95% CI	sig	SE	z	p
AC	Antisaccade	0.699	0.606 — 0.792	***	0.048	14.699	0.000
AC	Flanker	−0.264	−0.354 — −0.173	***	0.046	−5.681	<0.001
AC	Stroop	−0.325	−0.412 — −0.238	***	0.044	−7.339	<0.001
VA_ns	VAorient_k	1.000	1.000 — 1.000		0.000
VA_s	VAorient_S_k	1.000	1.000 — 1.000		0.000
WMC	OSpan	0.682	0.632 — 0.732	***	0.026	26.608	0.000
WMC	RotSpan	0.779	0.739 — 0.818	***	0.020	38.477	0.000
WMC	RunDigit	0.656	0.603 — 0.710	***	0.027	24.136	0.000
WMC	RunSpatial	0.810	0.774 — 0.846	***	0.018	44.163	0.000
WMC	SymSpan	0.780	0.741 — 0.820	***	0.020	38.781	0.000
* p < .05; p < .01; * p < .001

Predictor	DV	Standardized
Regression Paths
Predictor	DV	β	95% CI	sig	SE	z	p
AC	VA_ns	0.190	−0.091 — 0.471		0.143	1.327	0.185
WMC	VA_ns	0.436	0.178 — 0.695	***	0.132	3.307	<0.001
AC	VA_s	0.432	0.095 — 0.768	*	0.172	2.513	0.012
WMC	VA_s	0.315	−0.011 — 0.641		0.166	1.897	0.058
* p < .05; p < .01; * p < .001

Factor	Factor	r	95% CI	sig	SE	p
Latent Factor Correlations
WMC	AC	0.838	0.728 — 0.948	***	0.056	0.000
VA_ns	VA_s	0.268	0.175 — 0.361	***	0.047	<0.001
* p < .05; p < .01; * p < .001

Factor	Variance	Std. Variance	sig	p
Latent Factor Variance/Residual Variance
VA_ns	0.633	0.635	***	0.000
VA_s	0.485	0.486	***	0.000

Variable	R²
Explained Variance
VA_ns	0.365
VA_s	0.514

Interpretation

The extent to which individual differences in visual arrays performance reflects attention control processes depends on the nature of the task

Non-selective visual arrays reflects individual differences in working memory capacity, more so than attention control
Selective visual arrays reflects individual differences in attention control, more so than working memory capacity

Your Turn!

1. Confirmatory Factor Analysis

Create latent factors

Factor1 =~ Task1 + Task2 + Task3

Factor2 =~ Task4 + Task5 + Task6

Create single-indicator factors

Factor3 =~ 1*Task7

Factors are correlated by default, but to manually specify

Factor1 ~~ Factor2

Factor1 ~~ Factor3

Factor2 ~~ Factor3

Fit the model

fit <- cfa(model, data, missing = "fiml", std.ov = TRUE)

Your Turn!

1. Confirmatory Factor Analysis

model <- '
# latent factors
WMC =~ OSpan + SymSpan + RotSpan + RunSpatial + RunDigit
AC =~ Antisaccade + Flanker + Stroop

# single-indicator factors
VA_ns =~ 1*VAorient_k
VA_s =~ 1*VAorient_S_k
'

fit <- cfa(model, data, missing = "fiml", std.ov = TRUE)

Your Turn!

1. Confirmatory Factor Analysis

Get output

summary(fit, fit.measures = TRUE, standardized = TRUE)

sem_tables(fit)

Evaluate model fit

Your Turn!

1. Confirmatory Factor Analysis

Get output

summary(fit, fit.measures = TRUE, standardized = TRUE)

sem_tables(fit)

Evaluate loadings

Statistically significant, p < .05
Medium to large in size (standardized loading), > .30
Fairly balanced across the factor

Evaluate factor correlations

Your Turn!

2. Structural equation model

Copy CFA model syntax

Set predictor factors variance to 1

Factor1 =~ NA*Task1 + Task2 + Task3

Factor1 ~~ 1*Factor1

Define regression paths

dv ~ Factor1 + Factor 2

Fit the model

fit <- sem(model, data, missing = "fiml", std.ov = TRUE)

Your Turn!

2. Structural equation model

model <- '
# latent factors
WMC =~ NA*OSpan + SymSpan + RotSpan + RunSpatial + RunDigit
AC =~ NA*Antisaccade + Flanker + Stroop

WMC ~~ 1*WMC
AC ~~ 1*AC

# single-indicator factors
VA_ns =~ 1*VAorient_k
VA_s =~ 1*VAorient_S_k

# regression paths
VA_ns ~ WMC + AC
VA_s ~ WMC + AC
'

fit <- sem(model, data, missing = "fiml", std.ov = TRUE)

Your Turn!

2. Structural equation model

Get output

summary(fit, fit.measures = TRUE, standardized = TRUE)

sem_tables(fit)

Evaluate model fit

Evaluate regression paths

Evaluate latent and residual factor correlations

Three common categories of structural equation models

[image]

Correlated predictor model

Standard multiple regression or hierarchical regression
Multiple predictor variables, One dependent variable
Type of research question:
- Does variable x1 account for unique variance in variable y1, after accounting for variable x2?
- What is the unique contribution of variables x1 and x2 to variance in variable y1?

Correlated predictor model

Martin et al. (2021)

Data Set 3 (N = 568)

The purpose of this study was to evaluate the extent to which selective and non-selective visual arrays tasks reflect working memory storage and maintenance vs. attention control processes.

Correlated predictor model

Martin et al. (2021)

Data Set 3 (N = 568)

What is the unique contribution of working memory capacity and attention control to variability in non-selective and selective visual arrays?

[image]

Common cause model

One or more predictor variables, Two or more dependent variable
Type of research question:
- Does variable x1 account for the correlation between variables y1 and y2?
- Is variable x1 a common source of variance between variables y1 and y2?

Can attention control account for the correlation between working memory capacity and visual arrays?

Common cause model

Mediation model

One or more predictor variables, one or more mediators, and one or more dependent variables
Type of research question:
- Does variable m1 mediate the correlation between variables x1 and y1?
- Does variable m1 account for part of the correlation between variables x1 and y1?
Very similar to common cause model but can give you some additional information
- A precise statistical test of whether the mediator partially explains the co-variance between two variables
- Can make claims about full or just partial mediation

Does attention control mediate the relationship between working memory capacity and visual arrays?

Mediation model